OLS ESTIMATION AND THE t TEST REVISITED IN RANK‐SIZE RULE REGRESSION*

Abstract

The rank-size rule and Zipf's law for city sizes have been traditionally examined by means of OLS estimation and the t test. This paper studies the accurate and approximate properties of the OLS estimator and obtains the distribution of the t statistic under the assumption of Zipf's law (i.e., Pareto distribution). Indeed, we show that the t statistic explodes asymptotically even under the null, indicating that a mechanical application of the t test yields a serious type I error. To overcome this problem, critical regions of the t test are constructed to test the Zipf's law. Using these corrected critical regions, we can conclude that our results are in favor of the Zipf's law for many more countries than in the previous researches such as Rosen and Resnick (1980) or Soo (2005). By using the same database as that used in Soo (2005), we demonstrate that the Zipf law is rejected for only one of 24 countries under our test whereas it is rejected for 23 of 24 countries under the usual t test. We also propose a more efficient estimation procedure and provide empirical applications of the theory for some countries.

Zipf's Law in the Case of Moroccan Cities

“…For a zipfian urban system, the slope α should be practically equal to ‐1; in this case we say that the rank‐size rule is validated. The t ‐test is used to decide if the value of the exponent α is equal to ‐1 or not (Nishiyama et al ., , p. 692).…”

Historical urban growth in Europe (1300–1800)

“…The main problem is that the maximum likelihood (ML) estimator is more efficient if the underlying stochastic process is really a Pareto distribution (Gabaix & Ioannides, 2004;Goldstein, Morris, & Yen, 2004). Furthermore, as Gabaix and Ioannides (2004), Nishiyama, Osada, and Sato (2008), and Clauset et al (2009) emphasize, the estimates of the Pareto exponent are subject to systematic and potentially large errors. Finally, Gabaix and Ibragimov (2011) point out that this procedure is strongly biased in small samples.…”

Are Eu Countries Less Integrated Than U.S. States? Theory and Evidence*

“…Given these estimates, evidence against Zipf's law is then assessed by testing if an estimated slope coefficient is significantly different from −1. However, it has been shown that the OLS estimate of β j in and its associated standard error are biased downward, with the bias diminishing as the number of observational units ( M ) increases (Gabaix and Ioannides, 2004; Nishiyama and Osada, 2004; Nishiyama, Osada, and Sato, 2008). Hence, failure to correct for these biases means one would more often reject Zipf's law when it is in fact true.…”